What is the minimum k for which there exists a circle centered at (0,0) that is tangent to Sin(kx)?

Question

For y = Sin(x), there is no circle with origin (0,0) that can be tangent to the Sin(x) function anywhere locally. For a certain k and greater, a circle with origin (0,0) can be tangent to the Sin(kx) function. What is the minimum value k for which such a circle exists, and what is the diameter of the circle?

The circle with the embedded tangent Sin(kx) looks like the Yin-Yang symbol, without the dots.

Yeah, that's right folks, ksoileau nailed this one. Right down to 5th place accuracy. What was his method of numerical approximation?

Answer 1

Let (x,y) be the first nontrivial tangency point for minimal k.. Since it lies on a circle, the tangent vector is a scalar multiple of (1,-x/y). Since the circle is tangent to sin(kx) at (x,y), and the tangent vector to sin(kx) at (x,y) is a scalar multiple of (1,k*cos(k*x)), we must have k*cos(k*x)=-x/y. But y=sin(k*x), so we must have k*cos(k*x)=-x/sin(k*x), i.e., k*sin(k*x)*cos(k*x)+x=0. Thus the k we seek is the smallest positive k such that k*sin(k*x)*cos(k*x)+x=0 has a solution such that k*x is in the second quadrant. Through numerical approximation, I get that k is approximately 2.14554, with x approximately 1.0471, making y approximately .780208, thus the radius of the circle is approximately 1.30581.

EDIT: To find k, substitute x=z/2/k in the equation k*sin(k*x)*cos(k*x)+x=0. This implies 1/k^2=-sin(z)/z. To find the smallest possible k, find the absolute maximum for positive z of the function -sin(z)/z, and use this maximum value to solve for k. Then use x=z/2/k to solve for x, then y=sin(k*x)=sin(z/2) to solve for y. Finally, the radius r of the circle satisfies r^2=x^2+y^2.

Answer 2

10 points to ksoileau for a neat solution to an interesting
problem. I too, got the same equation by equating derivatives
of y = sin(kx) and y = sqrt(r^2 - x^2) (top half of circle), but
after that, I was stumped, but I really would like to know.
Can either of you, or anyone, briefly describe the numerical
approximation method that leads to a minimal k? Does it
have a specific name so I can look it up, to save describing
it? This is something new to me that I would like to learn.